- Each grid is 1 whole split into 100 equal cells, so each cell is one hundredth: 1/100, 0.01 or 1%.
- Fill rows: tap a cell and every cell up to it fills (tap the last one again to take it off, or slide along). Fill columns fills down the columns instead. Paint: tap or drag over any cells, so 1/4 can be a 5 × 5 square.
- 10 25 50 75 100 fill that many, 🎲 a random number, and Wholes + adds a full square beside the grid for values more than 1 (up to 3). Or type 3/5, 1 1/4, 0.35 or 35% and press Shade; something like 1/3 shades the nearest hundredth and says why it can’t be exact.
- ⇅ Tidy keeps the same number of cells but puts them in order. ◑ Swap swaps shaded and unshaded.
- Under the board each grid has its values: hundredths, fraction, simplest form, decimal, percentage, and tenths and hundredths. With 🙈 on (the usual) they stay hatched until you tap them, so the class can predict first. Tap a column heading or 👁 Reveal all for the lot.
- With two grids, the circle between them shows <, = or > (tap it), and Comparing gives A + B and the difference.
- Tenths lines, Running totals (10, 20, 30 …) and 1–100 help with counting. Not shaded shows the rest of the grid too.
- ▦ Explore: one or two grids to shade freely. 🎯 Shade it: shade the target, then Check. ✏ Name it: the grid is shaded for you; type what it shows as a fraction, decimal or percentage. Four levels, from tenths and quarters to more than 1. Each mode keeps its own work.
Shortcuts only work after you’ve clicked on the tool, so they never get in the way of the rest of the page.
Press 📋 Questions (or Q) to pick a question, even in full screen. Set it up loads it (↩ brings back what was there). 🖨 prints this board, a worksheet at the current level (six grids to name, six to shade, answers on the last page) or six blank grids. 🔒 stops taps changing the grids while you project. 🔗 copies a link that opens the tool as it is now. Nothing is sent anywhere, and nothing is stored on this device.
Investigation questions
62 questions. Work through these with the tool above. Set it up puts the grids on the board for you (↩ undo brings back what was there), and Show on board puts the question on screen, full size.
First experiments
The grid shows 1 shaded cell. What part of the whole square is one cell? Predict the Fraction, Decimal and Percentage, then tap each hatched value to check.
Teacher notes
1/100 = 0.01 = 1%. Every cell is one hundredth, which is why this grid links all three: the number of cells is the numerator, the hundredths and the percentage at once.
Tap the 4th cell of the third row. How many cells fill? Predict the decimal before you reveal it.
Teacher notes
24 cells: 24/100 = 6/25 = 0.24 = 24%. Fill rows shades every cell up to the one you tap, in reading order, so two full rows and 4 more.
Press 25, then 50, then 75. Each time, predict the Simplest form before you reveal it.
Teacher notes
1/4, 1/2 and 3/4 (0.25, 0.5 and 0.75). Ask why 50 hundredths is written 0.5 rather than 0.50: 5 tenths and 50 hundredths are the same amount.
Tenths lines are on and the grid shows 50 cells. How many tenths are shaded? Why does the decimal say 0.5 and not 0.50?
Teacher notes
5 tenths, which is 50 hundredths. 0.5 = 0.50: a zero in the hundredths place adds nothing, so the decimal is written without it.
Press 🎲. Before revealing anything, estimate the percentage shaded to the nearest 10%. Running totals are on to help. Then reveal the Percentage.
Teacher notes
🎲 shades 1 to 100 cells, each equally likely. The running total at the last full row gives the tens straight away, so most estimates land within 10% of the real value.
Tenths and hundredths
The grid shows 3 full rows. Predict the decimal. Then tap to add 4 more cells and predict again. What does Tenths and hundredths say each time?
Teacher notes
0.3, then 0.34. 30 cells is 3 tenths and 0 hundredths; 34 cells is 3 tenths and 4 hundredths, which is why the 3 and the 4 sit in those places.
Grid A shows 0.07. A student says it is the same as 0.7. Shade 0.7 on Grid B to show them the difference.
Teacher notes
0.7 is 70 cells: 7 full rows. 0.07 is 7 cells, less than one row. 0.7 is ten times 0.07.
Fill columns is on, with Tenths lines. Press 50. Each tenth is now a column. Does the decimal change when the tenths are columns instead of rows?
Teacher notes
No: 50 cells is 0.5 whichever way they are arranged. The tenths lines follow the fill direction, so a tenth can be a row or a column.
Reveal Tenths and hundredths for 0.35. Predict what it will say for 0.53 and 0.05, then type each into the box and press Shade to check.
Teacher notes
0.35 is 3 tenths and 5 hundredths, 0.53 is 5 tenths and 3 hundredths, 0.05 is 0 tenths and 5 hundredths. 0.35 and 0.53 use the same digits but differ by 18 hundredths.
Grid A shows 0.4 and Grid B shows 0.38. Many students say 0.38 is bigger because 38 is bigger than 4. Predict the sign in the circle, then tap it.
Teacher notes
0.4 > 0.38: 40 hundredths against 38. Writing 0.4 as 0.40 makes the comparison fair. Watch for “longer is larger”.
The grid shows 0.1 with 1–100 on. Now shade 0.01. How many times bigger is 0.1 than 0.01?
Teacher notes
10 times: 0.1 is 10 cells and 0.01 is 1 cell. Each tenth is ten hundredths.
Fractions on the grid
Press 10. Predict the Fraction and the Simplest form, then reveal both.
Teacher notes
10/100 = 1/10. One row is one tenth of the square, so the picture and the simplest form say the same thing.
Type 1/5 and press Shade. How many cells? Predict the number of cells for 2/5, 3/5 and 4/5, then type each to check.
Teacher notes
20, 40, 60 and 80 cells. 1/5 = 20/100 because 100 ÷ 5 = 20, so each fifth is 20 cells.
This grid shows 35 hundredths. Predict its Simplest form. How do you know it can’t be simplified any further?
Teacher notes
7/20. The highest common factor of 35 and 100 is 5; 7 and 20 share no factor except 1.
Grid A shows 36 cells. Predict whether it simplifies, then reveal. Now shade 37 cells. Does that simplify?
Teacher notes
36/100 = 9/25 (dividing by 4). 37/100 is already in its simplest form: 37 is prime and doesn’t divide 100. A number of cells only simplifies if it shares a factor 2 or 5 with 100.
How many numbers of cells from 1 to 99 give a Simplest form with denominator 20? The grid starts at 5. Predict, then test a few.
Teacher notes
8 of them: 5, 15, 35, 45, 55, 65, 85 and 95. They are the odd multiples of 5 that aren’t multiples of 25, so the highest common factor with 100 is exactly 5.
The grid shows 3/4. Press ◑ Swap. What fraction is shaded now? Not shaded is on: check both columns.
Teacher notes
After the swap, 1/4 is shaded and 3/4 isn’t. A fraction and the part left over always add to 1 whole.
Percentages
Shade 1 cell, then 10 cells, then 100 cells, revealing the Percentage each time. Why does the number of cells always equal the percentage on this grid?
Teacher notes
1%, 10% and 100%. Per cent means “out of 100” and the grid has 100 cells. On a grid of 50 cells, each cell would be 2%.
Shade 20%. Predict its Simplest form before you reveal it.
Teacher notes
20 cells: 20/100 = 1/5. Two rows out of ten, or one fifth.
Grid A shows 45% in rows. Use Paint to shade 45% of Grid B in a completely different shape. Do any of the values change?
Teacher notes
No: any 45 cells is 45% = 0.45 = 9/20. The circle shows = once Grid B has 45 cells.
What percentage is 3/20? Predict, then type 3/20, press Shade and reveal the Percentage.
Teacher notes
15%: 3/20 = 15/100, multiplying top and bottom by 5.
The grid shows 63% with Not shaded on. Predict the percentage that isn’t shaded, then reveal both columns.
Teacher notes
37%. The shaded and unshaded percentages always add to 100%, so 0.63 + 0.37 = 1.
One amount, three names
Grid A shows 0.6. Write it as a fraction in its simplest form and as a percentage, then reveal to check.
Teacher notes
3/5 and 60%. 60/100 simplifies by 20.
Type 2/25 and press Shade. Predict the decimal first. Some people say 0.225 or 0.0225: what does the grid say?
Teacher notes
8 cells: 0.08 = 8%. 2/25 = 8/100 because 25 × 4 = 100. Writing the numerator and denominator side by side as a decimal is a common error.
Put these in order without the tool: 0.3, 1/4, 28%, 3/10. Then shade each in turn and use the Hundredths value to check.
Teacher notes
1/4 (25) < 28% (28) < 0.3 = 3/10 (30). Turning everything into hundredths makes the order a count.
Everything for 0.05 is revealed. Predict every value for 0.5, then press 50. Which values are 10 times bigger, and which changed in a different way?
Teacher notes
Hundredths, decimal and percentage are all 10 times bigger (5 → 50, 0.05 → 0.5, 5% → 50%). The simplest form goes from 1/20 to 1/2: the denominator is divided by 10 instead.
Which is the odd one out: 0.45, 9/20, 45% or 4.5%? Shade the first three to check, then try typing 4.5%.
Teacher notes
4.5% is the odd one out: it is 0.045, four and a half hundredths, so it can’t be shaded exactly. The others are all 45 cells.
Same amount, different shapes
Grid A shows a quarter as 25 cells in rows. Grid B shows a quarter as a 5 × 5 square. Predict the sign in the circle, then tap it.
Teacher notes
=: both are 25 hundredths. The shape doesn’t change the amount; only the number of equal parts shaded does.
Use Paint to shade a quarter of the grid in a way nobody else in the class will think of. Check with Simplest form, then press ⇅ Tidy.
Teacher notes
Any 25 cells is 1/4. ⇅ Tidy keeps the 25 cells but puts them in order, which shows they fill two and a half rows.
The grid shows a 4 × 5 rectangle. Without counting every cell, what decimal is it? Then press ⇅ Tidy.
Teacher notes
20 cells (4 × 5) = 0.2 = 1/5. Tidy puts them into exactly 2 rows: 2 tenths.
With Paint, shade every cell around the edge of the grid. Predict the percentage before you reveal it.
Teacher notes
36 cells = 36%. Four sides of 10 counts each corner twice, so 40 − 4 = 36.
Shade one diagonal from corner to corner, then both diagonals. What fraction is each?
Teacher notes
One diagonal is 10 cells = 1/10. Both are 20 cells = 1/5: on a 10 × 10 grid the diagonals cross at a corner point, not a cell, so no cell is counted twice.
Comparing two grids
Grid A shows 3/5 and Grid B shows 13/20. Which is bigger? Predict, then tap the circle and reveal the difference.
Teacher notes
13/20 > 3/5: 65 against 60 hundredths, a difference of 5/100 = 1/20 = 0.05.
Grid A shows 40 cells. Shade Grid B so that A + B is exactly 1. Reveal A + B to check.
Teacher notes
Grid B needs 60 cells: 0.6 = 3/5 = 60%. Pairs that make 1 are pairs that make 100 hundredths.
Grid A shows 0.75 and Grid B shows 0.5. Predict the difference as a fraction and as a decimal, then reveal it.
Teacher notes
A − B = 0.25 = 1/4 = 25%. Count the extra 25 cells on Grid A.
Paint is on. Make the circle show = with two grids that look completely different.
Teacher notes
Any two shadings with the same number of cells are equal, however they look. A good check that students are comparing amounts, not shapes.
Grid A shows 1/4 and Grid B shows 1/5. A student says 1/5 is bigger because 5 is bigger than 4. Predict the sign, then tap the circle.
Teacher notes
1/4 > 1/5: 25 cells against 20. More equal parts means smaller parts.
Grid A shows 0.45 and Grid B shows 54%. Predict A + B and the difference, then reveal them.
Teacher notes
A + B = 0.99 = 99/100 = 99%. The difference is B − A = 0.09 = 9/100 = 9%. Swapping the digits changed the value by 9 hundredths.
More than 1
Grid A is full. Predict the fraction, decimal and percentage. Then press Wholes + and predict again.
Teacher notes
A full grid is 100/100 = 1 = 100%. With one extra whole it is 200/100 = 2 = 200%.
This shows 1 whole and 35 hundredths. Predict the Fraction, Simplest form, Decimal and Percentage, then reveal them.
Teacher notes
135/100 = 1 35/100, which simplifies to 1 7/20 = 27/20. As a decimal 1.35, as a percentage 135%.
Type 1 1/4 and press Shade. Then type 5/4. What do you notice?
Teacher notes
Both shade 1 whole and 25 cells: 125 hundredths = 1.25 = 125%. 5/4 and 1 1/4 are the same number written two ways.
Shade 150%. A student says a percentage can’t be more than 100. What does the grid show?
Teacher notes
1 whole and 50 cells: 1.5 = 3/2 = 1 1/2. More than 100% means more than the whole, for example a price that has gone up by a half.
This shows 2.05. Predict Tenths and hundredths, then reveal it. Why is the 0 needed?
Teacher notes
2 ones, 0 tenths and 5 hundredths. Without the 0, 2.5 would be 2 ones and 5 tenths, which is 250 hundredths.
Fractions that don't fit
Type 1/3 and press Shade. Why can’t the grid show it exactly? Which shading is closest?
Teacher notes
1/3 is 33 1/3 hundredths, so the tool shades 33 (0.33) and says 34 would be 0.34. As a decimal 1/3 = 0.333… recurring, which never ends.
Now try 1/8. What does the message say? What is 1/8 as a decimal?
Teacher notes
1/8 is 12 1/2 hundredths, exactly halfway, so the tool shades 13 and says 12 would be 0.12. 1/8 = 0.125 = 12.5%.
Which of these fit exactly on the grid: 1/2, 1/3, 1/4, 1/5, 1/6, 1/8, 1/10? Predict, then type each.
Teacher notes
1/2, 1/4, 1/5 and 1/10 fit (50, 25, 20 and 10 cells). 1/3, 1/6 and 1/8 don’t. A unit fraction fits only if its denominator divides 100.
Many people say 2/3 is about 67%. Type 2/3. How far out is 67 cells?
Teacher notes
2/3 is 66 2/3 hundredths, so 67 cells is 1/3 of a hundredth too much. 2/3 = 0.666… recurring, which rounds to 0.67.
Shade it and Name it
In 🎯 Shade it, shade 7/20 and press Check ✓. How did you work out the number of cells?
Teacher notes
35 cells. 100 ÷ 20 = 5 cells for each twentieth, and 7 × 5 = 35.
Shade 0.08 and press Check ✓. Then press 🎲 New target a few times at level 3. Which targets are you most likely to get wrong?
Teacher notes
8 cells. Place value targets (0.08 against 0.8) and fractions that aren’t in their simplest form are the usual traps. The feedback says when a shading is 10 times out.
In ✏ Name it, write the shaded part as a decimal and press Check ✓.
Teacher notes
0.45: 4 full rows (4 tenths) and 5 more cells (5 hundredths).
Write the shaded part as a fraction in its simplest form. What happens if you type 24/100?
Teacher notes
6/25. 24/100 gets “Nearly”: it is right but both numbers still divide by 4.
Level 4: write the shaded part as a percentage.
Teacher notes
175%: 1 whole is 100% and the 75 cells are 75%. As a decimal 1.75, as a fraction 1 3/4.
Misconceptions
A student says 0.5 is smaller than 0.25 because 5 is smaller than 25. Grid A shows 0.5 and Grid B shows 0.25. Predict the sign, then tap the circle.
Teacher notes
0.5 > 0.25: 50 hundredths against 25. The student is reading the digits after the point as whole numbers.
A student writes 3/10 as 0.03. Grid A shows 3/10 and Grid B shows 0.03, with Tenths lines on. What does each look like?
Teacher notes
3/10 is 3 full rows (0.3); 0.03 is 3 cells (3/100). 3/10 is ten times 0.03.
“25% means 1/25.” Type 1/25 and press Shade. Is the student right?
Teacher notes
No: 1/25 is 4 cells, which is 4%. 25% is 25 cells, which is 1/4.
A student shades 3 cells for 3/4. Grid A shows 3/4 and Grid B shows what they shaded. How would you explain the difference?
Teacher notes
3/4 is 75 cells; 3 cells is 3/100. The 3 in 3/4 counts parts of a whole split into 4 equal parts, not cells.
“0.20 is bigger than 0.2 because it has more digits.” Type 0.20 on Grid A and 0.2 on Grid B, then tap the circle.
Teacher notes
Both are 20 cells, so the circle shows =. A zero at the end of a decimal adds nothing.
A student says 1/10 = 10%, so 1/5 = 5%. Shade 1/5 and reveal its Percentage.
Teacher notes
1/5 = 20%. The 1/10 = 10% match is a coincidence of 10 × 10 = 100; it isn’t a rule.
Open investigations
Find every number of cells from 1 to 100 whose simplest form has a numerator of 1. How many are there?
Teacher notes
9: 1, 2, 4, 5, 10, 20, 25, 50 and 100 cells, giving 1/100, 1/50, 1/25, 1/20, 1/10, 1/5, 1/4, 1/2 and 1. They are the factors of 100.
Which numbers of cells from 1 to 100 give a simplest form with an odd denominator? What do they have in common?
Teacher notes
The multiples of 4 (25 numbers): 4/100 = 1/25, 20/100 = 1/5, 100/100 = 1. Dividing out a factor of 4 leaves 25, 5 or 1 on the bottom.
Make A + B exactly 1.5 using two grids that each show less than 1. How many different pairs can you find?
Teacher notes
Grid A from 51 to 99 cells, with B = 150 − A: 49 pairs, or 25 if A and B the other way round count as the same.
Print ▦ Six blank grids with answer boxes. Shade a different amount on each, then swap with a partner who writes each one as a fraction, a decimal and a percentage.
Teacher notes
An open task. Ask students to include one amount over 1 (drawing an extra whole) and one that tests place value, such as 0.06.
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